Obstructions to Erdös-Pósa Dualities for Minors

Let$\mathcal{G}$and$\mathcal{H}$be minor-closed graph classes. We say that the pair$(\mathcal{H},\ \mathcal{G})$is an Erdös-Pósa pair (EP-pair) if there exists a function$f$such that for every$k$and every graph$G\in \mathcal{G}$, either$G$has$k$pairwise vertex-disjoint sub graphs which do not belong to$\mathcal{H}$, or there exists a set$S\subseteq V(G)$of size at most$f(k)$for which$G-S\in \mathcal{H}$. The classic result of Erdös and Pósa says that if$\mathcal{F}$is the class of forests, then$(\mathcal{F}, \mathcal{G})$is an EP-pair for all graph classes$\mathcal{G}$. A minor-closed graph class$\mathcal{G}$is an EP-counterexample for$\mathcal{H}$if$\mathcal{G}$is minimal with the property that$(\mathcal{H},\ \mathcal{G})$is not an EP-pair. In this paper, we prove that for every minor-closed graph class$\mathcal{H}$the set$\mathfrak{C}_{\mathcal{H}}$of all EP-counterexamples for$\mathcal{H}$is finite. In particular, we provide a complete characterization of$\mathfrak{C}_{\mathcal{H}}$for every$\mathcal{H}$and give a constructive upper bound on its size. We show that each class$\mathcal{G}$in$\mathfrak{C}_{\mathcal{H}}$can be described as the set of all minors of some, suitably defined, sequence of grid-like graphs$\langle{W}_{k}\rangle_{k\in \mathbb{N}}$. Moreover, each$\mathrm{W}_{k}$admits a half-integral packing, i.e.,$k$copies of some$H\not\in \mathcal{H}$where no vertex is used more than twice. This implies a complete delineation of the half-integrality threshold of the Erdös-Pósa property for minors and as a corollary, we obtain a constructive proof of Thomas' conjecture on the half-integral Erdös-Pósa property for minors which was recently confirmed by Liu. Our results are algorithmic. Let$h=h(\mathcal{H})$denote the maximum size of an obstruction to$\mathcal{H}$. For every minor-closed graph class$\mathcal{H}$, we construct an algorithm that, given a graph$G$and an integer$k$, either outputs a half-integral packing of$k$copies of some$H\not\in \mathcal{H}$or outputs a set of at most$2^{k^{\overline{\mathcal{O}}_{h}(1)}}$vertices whose deletion creates a graph in$\mathcal{H}$in time$2^{2^{k^{\mathcal{O}_{h}(1)}}}\cdot\vert G\vert ^{4}\log\vert G\vert$. Moreover, as a consequence of our results, for every minor-closed class$\mathcal{H}$, we obtain min-max-dualities, which may be seen as analogues of the celebrated Grid Theorem of Robertson and Seymour, for the recently introduced parameters$\mathcal{H}$-treewidth and elimination distance to$\mathcal{H}$.